Dynamic interaction of suspension-type monorail vehicle and bridge: Numerical simulation and experiment

Mechanical Systems and Signal Processing 118 (2019) 388–407

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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

Dynamic interaction of suspension-type monorail vehicle and bridge: Numerical simulation and experiment Chengbiao Cai, Qinglie He, Shengyang Zhu ⇑, Wanming Zhai ⇑, Mingze Wang Train and Track Research Institute, State Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu 610031, China

a r t i c l e

i n f o

Article history: Received 12 June 2018 Received in revised form 13 August 2018 Accepted 29 August 2018

Keywords: Suspension-type monorail Vehicle-bridge interaction APDL Mixed integration method Multibody dynamics Finite element theory Field test

a b s t r a c t To evaluate the dynamic behaviour of suspension-type monorail (STM) system, a coupled dynamic model of the STM vehicle-bridge system has been developed based on multibody dynamics and finite element (FE) theory. In the model, both the spatial vehicle model with 21 degrees of freedom and the 3-dimensional FE model of the bridge structure for a particular STM system are established by using ANSYS parametric design language (APDL). The vehicle subsystem is coupled with the bridge subsystem through the contact relation between the vehicle tire and bridge inner surface. Then, a mixed explicit and implicit integration method is used to solve the coupled dynamic model. Finally, the dynamic responses of the vehicle-bridge system are calculated by adopting the established model, which are compared with the field test data. Results show that the simulation with the proposed dynamic model is in good agreement with the filed test data. Some apparent discrepancies can be distinguished when the bridge is treated as rigid body and flexible body, respectively, showing the importance of considering the flexible bridge and demonstrating several modelling advantages of the proposed coupled dynamic model. Overall, the train has good operation stability, and the lateral stability of car body is worse than its vertical stability because of the special vehicle structure. The first-order natural frequencies of the bridge in the vertical and lateral-torsional directions are about 5.60 Hz and 2.27 Hz, respectively. The bridge lateral acceleration is significantly larger than the vertical accelerations at the bridge middle-span section due to the low lateral stiffness of the bridge. These conclusions could provide a useful guidance for design of the STM system. Ó 2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

1. Introduction With the recent fast development of urban traffic, many types of the urban rail transit systems have been employed to solve the serious problem of traffic jam [1-4]. The suspension-type monorail (STM) transport system is an active technological solution, which has the advantages of stronger climbing ability, lower turning radius, lower running noise, lower cost and shorter construction period compared with subway and light railway [5,6]. It not only can be used as a supplement of the large-scale urban rail transit system, but also can be used as a main transportation power for small and mediumsized cities. The earliest representative countries in the world that studied the STM system and put it into operation are Germany and Japan, and they have accumulated systematic technology and experience through its continuous development and improvement. In recent year, China successfully completed the first test line for the STM system with the new energy of high ⇑ Corresponding authors. E-mail addresses: [email protected] (S. Zhu), [email protected] (W. Zhai). https://doi.org/10.1016/j.ymssp.2018.08.062 0888-3270/Ó 2018 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

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capacity lithium battery under independent research and innovation. Due to its natural mechanism, the STM system could also be considered as one of coupled vehicle-bridge systems, which has special vehicle structure, bridge structure and steering system employing rubber tires. To ensure the health and reliability of the STM transit system, it is necessary to systematically research the running safety and ride comfort of the vehicle, and the dynamic behavior of the bridge structure should also be evaluated. Considering the general concept of railway vehicle-bridge interaction, a number of literatures can be found [7–20]. In contrast with the conventional railway bridge, a limited number of studies have been carried out for monorail system which utilizes a different interaction mechanism with the bridge structure. Boehm and Frisch [5] summarized the composition, the operation mode and the main feature of the sky train in German Dortmund in detail. Goda [21] studied a curving dynamics for a straddle-type monorail car based on multi-body dynamics method. Lee et al. [22] studied the vehicle-bridge interaction of the straddle type monorail system based on numerical simulation and the ride comfort of running vehicle is estimates using 1/3 octave band spectral analysis. Naeimi et al. [23] investigated the coupled vibration characteristics of the straddle type monorail vehicle-bridge by combining multibody dynamic (MBD) and finite element (FE) modelling, and further gives an overview about the dynamic forces and reactions that can appear in bridge structure due to the vehicle movement. It can be seen that the investigation objective of most existing studies focus on the straddle type monorail system. However, a limited number of literatures have been carried out for the dynamic behaviour of the STM system. Meisinger [24,25] developed a vehicle dynamic model with four degrees of freedom to research the lateral dynamic of STM train with periodic irregularities. Bao [26] investigated the coupled vehicle-bridge vibration of the STM system under different track irregularities based on a co-simulation method; the bridge model is condensed into a single super element to consider main low-order modes of the bridge, and it is input into the SIMPACK to realize the vehicle-bridge interaction analysis. However, high-frequency local vibrations of the STM bridge could be induced by local defects or irregularities. In order to obtain the accurate analysis results of the bridge local vibrations, high-order modes of the bridge should be considered in studying the vehicle-bridge interaction. Further, the strength check of the bridge is an essential work for the consideration of the bridge operational reliability, but it is difficult for the co-simulation method to obtain the local dynamic stress of the bridge structure. In addition, the current researches are also lack of validation by field tests. Actually, the STM bridge is the steel box-beam with an open bottom, whose fundamental natural frequencies in the vertical and lateral direction are in the low range. If the vibration frequencies of the bridge subsystem are close to the natural frequencies of the vehicle subsystem, the resonance or larger vibration of vehicle-bridge coupled system will be induced, thus reducing ride safety and comfort of the monorail vehicle. Moreover, the vehicle running part of the bridge is a steel plate structure, which will appear high-frequency vibration under the vehicle load. Therefore, it is significant for the STM system to develop a coupled vehicle-bridge dynamic model considering a full-mode FE model of the bridge. This paper proposes a framework based on computer simulation to study the vehicle-bridge dynamic interaction for the STM system. First, both the spatial vehicle model with 21 degrees of freedom and the 3D FE model of the bridge structure for a particular STM system are established based on ANSYS parametric design language (ADPL). The vehicle and bridge subsystems are coupled through the contact relation between tire and bridge inner surface. Further, the fast explicit integration method is used to solve simultaneous differential equations of the multibody model of vehicle subsystem, and the finite model of bridge is solved using an implicit integration method. On this basis, dynamic responses of the coupled vehiclebridge system are investigated and validated with the field test data. Finally, some useful conclusions are obtained, which provide an important insight for the design and safety assessment of vehicle-bridge interaction of the STM system.

2. Vehicle–bridge dynamic interaction model of STM system Fig. 1 presents a STM transport line in China, and it was built in 2016. Due to its natural mechanism, the STM system is a type of coupled vehicle-bridge system, which is composed of the bridge subsystem and the moving vehicle subsystem using the lithium battery to provide the driving force.

2.1. Vehicle model of STM system A STM vehicle has special structure, whose dynamic model is different from that of traditional railway vehicle. Figs. 2–4 show the dynamic model of the STM vehicle subsystem, which consists of two bolsters, two bogies, two central pins and a car body. Each bogie consists of four driving tires and four guiding tires, which is placed inside the box-beam bridge. The driving tires act as the bogie primary suspension and carry the load of the whole vehicle. The guiding tires provide the lateral guide. The bolster is supported on the bogie with the secondary suspensions, and the central pin is connected to the bolster by using a rotating hinge. Further, an articulated four-linkage mechanism is used to connect the central pin and the car body, which releases the lateral constraint of the car body to a certain degree and reduces lateral impact of the body. In addition, a lateral shock absorber (c1) and two rubber blocks (k3 and k4) are installed inside the four-linkage mechanism, which can play an important role in decreasing the lateral vibration of the car body and limiting the lateral displacement of the car body, respectively.

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25m

10.8m

Fig. 1. Overview of the STM system in China.

Fig. 2. The front view of STM system.

2.49m

1.5m Fig. 3. The top view of STM system.

2.1.1. Model assumptions The basic unit of a STM train model is the vehicle model. The battery car provides power to keep vehicle running in the the box-beam, which interacts with bridge and is modeled by exerting concentration force to bridge due to its small weight. In this work, the train is formed by two vehicles and two battery cars, and each vehicle submodel is established based on a lumped mass approach. It is assumed that: (1) The car bodies, bogie frames, central pins and bolsters are all rigid bodies with no elasticity.

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Fig. 4. The end view of STM system.

(2) The vehicle passes through the bridge at a constant speed without consideration of the stretching vibration of the vehicle and the longitudinal dynamic interaction between neighbouring vehicles. (3) The car body is symmetrical about its centre of mass at x-, y- and z-directions. As shown in Figs. 2–4, each vehicle is divided into one car body, two bogies, two central pins and two bolsters. It is worth pointing out that there is only 1 degree of freedom(DOF) rotating freely around the Z axis between the bolster and the center pin, so the bolster and the center pin (named the hanging beam)can be treated as a rigid body to analyze its dynamic behavior at other motion directions. Overall, the entire vehicle is considered as five multi-rigid bodies system. Both the car body and the bogie have five DOFs: the vertical displacement Z, the lateral displacement Y, the roll angle U, the yaw angle W and the pitch angle b, respectively. The hanging beam, neglecting the yaw angle and the pitch angle due to small influence, has three DOFs: the vertical displacement Z, the lateral displacement Y, the roll angle U. As a result, the total DOFs of the vehicle are 21 as shown in Table 1. The tires and the secondary suspensions are modelled as spring-damper elements. As shown in Table 1, the subscript symbol c, t, h, d and g represent the car body, the bogies, the hanging beams, the driving tires and the guiding tires, respectively. The subscript symbol i (i = 1, 2) represents the number of vehicle. The subscript symbol j (j = 1, 2) represents the number of the fore and rear bogies and the hanging beams. The subscript symbol k (k = 1, 2) represents the number of the driving tires and the guiding tires of the fore and rear each bogies.

2.1.2. Spatial coordinate transformation As is shown in Fig. 4, in order to solve the coupled dynamic model of the STM vehicle subsystem, the four-link mechanism composed of node A, node B, node C and node D need to be decoupled. The oblique links (AC and BD) are equivalent as the spring component(k1 and k2) with the large stiffness that can be determined according to structure and material parameters. Simultaneously based on the method of spatial coordinate transformation, an absolute coordinate system and two local coordinate systems of the car body and the hanging beam are defined as shown in Fig. 4. Thereby, the real-time node coor-

Table 1 Degree of freedom of STM vehicle dynamic model. Vehicle component

Lateral motion

Vertical motion

Roll motion

Yaw motion

Pitch motion

Car body Front bogie Rear bogie Front hanging beam Rear hanging beam

Yci Yti1 Yti2 Yhi1 Yhi2

Zci Zti1 Zti2 Zhi1 Zhi2

U U U U U

W W W

bci bti1 bti2 – –

ci ti1 ti2 hi1 hi2

– –

ci ti1 ti2

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dinates (node A, B, C, D) at the end of the spring component (k1 and k2) can be obtained. Further, the tension force of the spring component (FLij and FRij) can be obtained by calculating the stretch amount of the spring k1and k2. Similarly, the forces (FHij, FLbij and FRbij) of the other spring compents can also be obtained. O-XYZ is the absolute coordinate system. Oc-XcYcZc is the local coordinate system of the vehicle body, Oh-XhYhZh is the local coordinate system of the boom, and their origins are the mass center of rigid body. The spatial coordinate system vector is e [x, y, z]. The conversion relationship between the local coordinate the car body (or the hanging beams) and the absolute coordinate system is written as below:

2

xc

3

2

coswc

6 7 6 4 yc 5 ¼ 4 cos/c sinwc zc sin/c sinwc 2

3 2 xh coswh 6 7 6 ¼ y 4 h 5 4 cos/h sinwh zh sin/h sinwh

sinwc cos/c coswc sin/c coswc

32 3 x 76 7 sin/c 54 y 5 z cos/c

sinwh cos/h coswh sin/h coswh

0

32 3 x 76 7 sin/h 54 y 5 z cos/h

ð1Þ

0

ð2Þ

2.1.3. Equations of motion For each rigid body of vehicle subsystem, the equations governing the vertical, pitch motion, lateral motion, roll motion and yaw motion of the lumped masses can be entirely described using second order ordinary differential equation in the time domain. They are formulated by applying the D’Alemenbert principal, as shown in Eqs. (3)–(19), whose notations are explained in Table 2. The left and right lateral forces of the secondary suspension (i, j = 1–2)

F yLij ¼ K yLij Y tij  Y hij  Hbt /tij  lh1 /hij þ

L2c 2Rci

!

 ! L2 d 1 þ C yLij Y_ tij  Y_ hij  Hbt /_ tij  lh1 /_ hij þ c 2 dt Rci

ð3Þ

Table 2 Notations of vehicle model. Notations

Descriptions

Mc, Mh and Mt Icx, Icy and Icz Ihx, Ihy and Ihz Itx, Ity and Itz (KzLij and KzRij) and (CzLij and CzRij) (KyLij and KyRij) and (CyLij and CyRij) FzLij and FzRij FyLij and FyRij FLij and FRij PdLijk and PdRijk PgLijk and PgRijk FCLijk and FCRijk MCLijk and MCRijk FHij FLbij and FRbij 2Lc, 2Lt and 2Lg

Mass of car-body, hanging beam and bogie X-inertia, Y-inertia and Z-inertia of car-body X-inertia, Y-inertia and Z-inertia of hanging beam X-inertia, Y-inertia and Z-inertia of bogie Left and right vertical stiffness and damping of secondary suspension on the jh bogie of the ith vehicle

2 l3 and 2 l5 l1 and l2 l0 a and b l4 l6 l7 2dw and 2ds lh1 and lh2 Hcb Hbt and Htw hLij and hRij bLij and bRij vLij and vRij Rci, Rhij and Rtij V

Left and right lateral stiffness and damping of secondary suspension on the jth bogie of the ith vehicle Left and right vertical forces of secondary suspension on the jth bogie of the ith vehicle Left and right lateral forces of secondary suspension on the jth bogie of the ith vehicle Tension forces of oblique link AC and link BD under the jth bogie of the ith vehicle Left and right driving tire radial forces of the kth wheelset on the jth bogie of the ith vehicle Left and right guiding tire radial forces of the kth wheelset on the jth bogie of the ith vehicle Left and right driving tire lateral forces of the kth wheelset on the jth bogie of the ith vehicle Left and right driving tire self-align moment of the kth wheelset on the jth bogie of the ith vehicle Tension force of lateral shock absorber under the jth bogie of the ith vehicle Forces of left and right rubber blocks Longitudinal distance between bogies, Distance between driving tire sets in bogie and Distance between guiding tire sets in bogie Length of link CD and Length of link AB Vertical distance from rubber block to link CDand link AB, respective Vertical distance from link AB to link CD Ratio of l2 to l0 and ratio of l1 to l0 Vertical distance from body centroid to lateral shock absorber Vertical distance from body centroid to rubber block Vertical distance from hanging beam centroid to lateral shock absorber Lateral distance of tires of each driving wheelset and Lateral distance of secondary suspension Vertical distance from hanging beam centroid to above surface of the secondary suspension and link AB, respectively Vertical distance from body centroid to link CD Vertical distance from bogie centroid to below surface of the secondary suspension and guiding tire position, respectively Angle of link AC and axis Y, Angle of link BD and axis Y Angle of link AC and link CD, Angle of link BD and link CD Angle of link AB ans link AC, Angle of link AB ans link BD Curve radius of carbody, hanging beam and bogie Vehicle running speed

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F yRij ¼ K yRij Y tij  Y hij  Hbt /tij  lh1 /hij þ

L2c 2Rci

!

 ! L2 d 1 þ C yRij Y_ tij  Y_ hij  Hbt /_ tij  lh1 /_ hij þ c 2 dt Rci

393

ð4Þ

The left and right verticle force of the secondary suspension (i, j = 1–2)

    F zLij ¼ K zLij Z hij  Z tij þ ds /tij  ds /hij þ C zLij Z_ hij  Z_ tij þ ds /_ tij  ds /_ hij

ð5Þ

    F zRij ¼ K zRij Z hij  Z tij  ds /tij þ ds /hij þ C zRij Z_ hij  Z_ tij  ds /_ tij þ ds /_ hij

ð6Þ

(1) Equation of motion of car body (i, j = 1–2) Vertical motion:

Mc Z€ ci ¼ Mg 

2 X

ðF Lij sinhLij þ F Rij sinhRij Þ

ð7Þ

€ ¼ ðF Li1 sinhLi1 þ F Ri1 sinhRi1  F Li2 sinhLi2  F Ri2 sinhRi2 ÞLc Icy b ci

ð8Þ

j¼1

Pitch motion:

Lateral motion: 2

V Mc Y€ ci þ Rci

! ¼ F Li1 coshLi1 þ F Li2 coshLi2  F Ri1 coshRi1  F Ri2 coshRi2 þ

2 X

F Hij þ ð1  aÞF Lbij  ð1  aÞF Rbij

ð9Þ

j¼1

Roll motion:

€ ¼ ðF Li1 sinb þ F Li2 sinb  F Ri1 sinb  F Ri2 sinb Þl3 þ ðF Hi1 þ F Hi2 Þl4 Icx / ci Li1 Li2 Ri1 Ri2 þðF Li1 cosbLi1 þ F Li2 cosbLi2  F Ri1 cosbRi1  F Ri2 cosbRi2 ÞHcb  P2  P2 þ j¼1 F Lbij  F Rbij l6 þ j¼1 ðaF Rbij  aF Lbij ÞHcb

ð10Þ

Car body yaw motion:

   € þV d 1 ¼ ðF Li1 coshLi1 þ F Ri2 coshRi2  F Li2 coshLi2  F Ri1 coshRi1 ÞLc þ ðFHi1  FHi2 ÞLc þ ðFLbi1 þ FRbi2 Icz w ci dt Rci  FLbi2  FRbi1 Þð1  aÞLc

ð11Þ

(2) Equation of motion of hanging beam(i, j = 1–2) Vertical motion:

€ hij ¼ Mh g þ FLij sinðhLij Þ þ FRij ðsinhRij Þ  FzLij  FzRij Mh Z

ð12Þ

Lateral motion: 2       € hij þ V Þ ¼ FyLij þ FyRij  FLij cos hLij þ FRij cos hRij  FHij þ b FRbij  FLbij M h ðY Rhij

ð13Þ

Roll motion:

      € ¼ F Lij cos p - v Ihx / lh2  F zRij ðds þ /hij lh1 Þ  FzLij ðds  /hij lh1 Þ hij Lij  F Rij cos p - vRij          þ F Rij sin p - vRij  F Lij sin p - vLij l5 þ F Hij l7  b F Rbij  F Lbij lh2 þ F yLij þ F yRij lh1

ð14Þ

Equation of motion of bogie (i, j, k = 1–2) Vertical motion:

Mt Z€ tij ¼ F zLij þ F zRij 

2 X 

 PdLijk þ PdRijk þ Mt g

ð15Þ

k¼1

Pitch motion:

  € ¼ PdLij1 þ PdRij1  PdLij2  PdRij2 Lt Ity b tij

ð16Þ

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Lateral motion:

    V2 Mt ðY€ tij þ Þ ¼ F yLij  F yRij þ PgLij1 þ P gLij2  PgRij1  PgRij2 Rtij

ð17Þ

Roll motion:

€ ¼ ðF zRij  F zLij Þds þ ðFyLij þ FyRij ÞHbt þ ðPdLij1 þ PdLij2  PdRij1  PdRij2 Þdw  ðPgLij1 þ PgLij2  PgRij1  PgRij2 ÞHtw Itx / tij

ð18Þ

Yaw motion:

  2 X   € þV d 1 ¼ ðPgLij1 þ P gRij2  PgLij2  PgRij1 ÞLg  Itz w ðFCLijk þ FCRijk ÞLt  MCLij1  MCRij1 þ MCLij2 þ MCRij2 tij dt Rtij k¼1

ð19Þ

2.2. Contact model between vehicle tire and bridge inner surface The solid rubber tires are employed to drive and guide for the STM vehicle, whose deformation and mechanism when contacting with bridge are different with the traditional steel wheel-rail system. In addition, compared with common inflatable tire using in the road vehicle system [27], the solid rubber tires have smaller deformation and different mechanical properties under vehicle dynamic load. Currently, despite the lack of the solid tire test, some useful results of the dynamic performance of the monorail system were obtained by modeling the tire as the linear spring-damping element [28]. Therefore, the linear spring-damping element is adopted here to model the solid tire, and simultaneously the lateral characteristics of the tires are also taken into consideration in this work. 2.2.1. Radial force model of the tire The linear spring-damping element is used to model the solid tire. The normal contact force between tire and bridge inner surface can be explained as:

( FZ ¼

  kðZw  ðZr  Zr0 ÞÞ þ c V_ DZ Dr > 0 0

Dr  0

ð20Þ

where Dr denotes the radial compression amount of the tires; k and c are the stiffness and the damping coefficients, respectively; Zw is the displacement of the tire; Zr and Zr0 are the bridge displacement at the contact point between tire and bridge and bridge inner surface roughness, respectively; and V_ DZ is the change rate of the tire radial displacement. 2.2.2. Lateral force characteristics of the tire Some researches have proved that the lateral force and self-align moment of inflatable tire will be produced when the vehicle passes through a curve, which cause a lateral angle of tire [27,29]. It is worth pointing out that compared with common road vehicle, the turning radius of the STM vehicle is larger and the guiding tire will provide lateral force to help the STM vehicle pass the curve bridge, which make the driving tires of the STM vehicle produce the smaller lateral angle. When the lateral angle a of each tire of a bogie is less than 4°-5°, it can be considered that the lateral force Fc and self-align moment Mc are linearly related to the lateral angle a [29-32]. In this regard, Fc and Mc can be espressed as:

F c ¼ k5 a

ð21Þ

Mc ¼ k6 a

ð22Þ

a ¼ arctan

v u

where k5 is the lateral stiffness of the tire, k6 is the self-align stiffness of the tire; and longitudinal velocity of the tire center, respectively.

ð23Þ

v and u are the lateral velocity and

2.3. FE model of the bridge The STM bridge is the simply supported box-beam structure with an open bottom, and the longitudinal stiffeners are employed to strengthen the bridge stiffness. The pin shafts are used as connectors between the girder and the piers, and the girder is not continuous with a appropriate gap for each 25 m. In this study, a detailed 3D FE model of the bridge is established based on ADPL, as shown in Fig. 5, which mainly consists of girders, piers, stiffeners and connectors. Linear plate element of Shell181 with six DOFs at each node is applied for the bridge structure. The simple-supported constraints are applied between the girders and the piers. All the nodes at the bottom of piers are subjected to fixed constrains. The bridge structure parameters adopted in the work are listed in Table 3.

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Fig. 5. 3D FE Model of the bridge: the entire FE mesh and the cross section of the FE model.

Table 3 Main bridge structure parameter. Components

Parameters

Values

Entire guideway system

Density Poisson ratio Elastic modulus Span of the bridge Space of the stiffener Cross section(width  height) Height Cross section

7840 kg/m3 0.3 210 Gpa 25 m 1.6 m 0.78 m  1.1 m 10.8 m 0.8 m  0.8 m

Box beam

Pier

2.4. Solution method This section describes the solution method of the coupled vehicle-bridge dynamic model for the STM system, which is entirely based on the ANSYS platform. First, the motion equations for the vehicle subsystem and the interaction relation between tire and bridge are programmed into ANSYS by APDL. Meanwhile, the FE model of the bridge subsystem is also created using Shell181 element in ANSYS by APDL. In each time step of dynamic simulation, the Newmark implicit integration method is selected to calculate the FE model, which ensures the stability of the FE model calculation, and then the dynamic responses of the FE model are acquired through ANSYS solver. Simultaneously, in order to improve the computational efficiency, the explicit integration method proposed by Zhai [33] is adopted to solve the motion equations of each component of the vehicle, which is also incorporated in ANSYS. On this basis, the dynamic displacement and velocity at the contact point of the bridge and each tire can be obtained at each time step, and the current tire forces can be calculated based on the contact relation between the tire and bridge inner surface, as shown in the section 2.2. Consequently, the tire forces are transmitted to the vehicle motion equations and the FE model of bridge in the next time step. In this way, the coupled dynamic analysis of STM vehicle and bridge can be readily realized, and its scheme is presented in Fig. 6. In order to guarantee the calculation precision of the coupled system, the time steps of the two numerical integration methods are set to be the same value of 1  10-4 s. It is worth noting that this method provides an efficient way to consider much complicated bridge structure models with high nonlinearities owing to the powerful modeling and solving capacity of ANSYS. In addition, the method can also consider the full-mode FE model of bridge in ANSYS software, which is also helpful to study the local vibration and damage of bridge structure.

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Input monorail vehicle technique parameters and vehicle motion equations in ANSYS using APDL

Create FE model of the suspension-type monorail bridge using Shell181 elements in ANSYS

Initialization, t=0 Determine the vehicle position on the bridge, input bridge inner surface roughness Solve vehicle motion equations using Zhai method

Obtain previous dynamic displacements and velocities of bridge at element nodes solved by ANSYS

Determine contact geometry between tires and bridge inner surface

Calculate the track dynamic responses at contact points between tires and bridge inner surface based on interpolation theory

Calculate tire forces

Apply equivalent tire forces at element nodes, perform dynamic analysis in ANSYS

Calculate accelerations of vehicle components bases on vehicle equations

Obtain bridge dynamic responses

No, t=t+Δt

Check if monorail vehicle leave the bridge

Fig. 6. Scheme of coupled dynamic analysis of STM vehicle and bridge.

3. Model validation Firstly, to check the FE model of the bridge, the mode shapes and natural frequencies are calculated. Further, a field test of the natural frequencies of the bridge is carried out by using the VibRunner acquisition instrument made by German M + P Company. The signal acquisition points and the exciting points are first arranged on the bridge sections, then by applying a hammer impact on the pre-set exciting points, the test signals can be obtained by the VibRunner data acquisition system with the accelerometers attached at the acquisition points and the force sensor installed in the hammer. On the basis, the vibration frequencies of the bridge structure can be obtained through analysing the transfer functions between the exciting points and the signal acquisition points using post processing software of the VibRunner. Finally, the calculation results can be compared with test results to validate the effectiveness of the bridge FE model. In addition, the Video Gauge System (VGS) and accelerometers are used to measure the dynamic displacements of the bridge and the accelerations of the vehicle and bridge subsystems, respectively. Fig. 7 shows the detailed measurement points of the dynamic displacements at the bridge middle-span and pier sections.

500

1525

1100 655

1544

1480

Observation point A

A

780 800 B

A

A

8900

240

800

1100

B

10800

25000 15×1600 Observation point

500

B

800

B

Unit: mm Fig. 7. Test points of the suspension monorail bridge.

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The VGS with a video camera is a contactless mobile device for measuring displacements of target points, as shown in Fig. 8. The main parts of the VGS include the video camera equipped with a telescope that captures the target motion, as well as the post-processing operation software which can identify the target displacements. This non-contact technology also allows for multi-point measurements of dynamic displacements. In the normal monitoring condition, the accuracy of 0.02 mm can be achieved within a distance of 10 m and the maximum sampling frequency is up to 117 Hz. The vibration accelerometers (version: 3100D24), produced by American Dytran company, are used to capture vibration accelerations of car body and bridge. The measurement range and the sensitivity of accelerometer are 0–5 g and 1026 mv/g, respectively. The accelerometers are mounted at the interior of the car body and the middle-span of the bridge, as shown in Fig. 9. 3.1. Comparison of bridge modes Figs. 10 and 11 show the first-order mode shapes of the bridge obtained by numerical simulation and field test, respectively. Comparison results of the natural frequencies of the bridge are presented in Table 5. As shown in Table 5, the first-two-order vertical bending modes of the bridge are 5.6 Hz and 15.19 Hz, respectively. It should be noted that the modes of the bridge are coupled in lateral and torsional directions. The first-two-order coupled lateral-and-torsional modes are 2.27 Hz and 9.55 Hz. The calculation results are in good agreement with the test results, showing the reliability of the bridge FE model. In addition, the damping ratio of the first vertical bending mode of the bridge is obtained by field test, which is about 1.7%. 3.2. Dynamic response comparisons of coupled vehicle-bridge system 3.2.1. Comparison of the dynamic responses in time-domain To validate the effectiveness of the established coupled model, a coupled model of the vehicle and straight bridges with length 150 m is then developed based on ADPL, whose dynamic responses are compared with those of the test data. In the dynamic simulation, the sample of surface roughness is obtained by field test, as shown in Fig. 12, and the main calculation parameters adopted in the work are listed in Tables 3 and 4.

(a)

(b)

Video camera

Measurement point at the top of pier Measurement point at the middle of bridge

Electronic theodolite

Computer (acquisition system)

Video Gauge System

Tripod Fig. 8. Dynamic displacement test of bridge: (a) Video Gauge System (VGS); (b) The site test chart.

(a)

Acceleration

(b)

Accelerometer

Fig. 9. Vibration acceleration test (a) The acceleration measurement point of car body; (b) The acceleration measurement point of bridge middle-span.

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Fig. 10. Calculation results of the bridge mode: (a) First-order vertical bending mode; (b) First-order coupled lateral-and-torsional mode.

Fig. 11. Test results of the bridge mode: (a) First-order vertical bending mode; (b) First-order coupled lateral-and-torsional mode.

16

Left driving surface Right driving surface

(a)

Left guiding surface Right guiding surface

Surface roughness (mm)

Surface roughness (mm)

60

40

20

0

-20

(b)

8

0

-8

-16

0

60

120

Distance (m)

180

240

0

60

120

180

Distance (m)

Fig. 12. Samples of bridge inner surface roughness: (a) Driving bridge surface; (b) Guiding bridge surface.

240

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C. Cai et al. / Mechanical Systems and Signal Processing 118 (2019) 388–407 Table 4 The main vehicle calculation parameter. Parameter

Units

Value (Empty car/Normal loading)

Car body mass Bogie mass Radial stiffness of the driving tire Lateral stiffness of the tire Radial stiffness of the guiding tire Radial damping of the driving tire Radial damping of the guiding tire Damping of lateral vibration absorber Distance between bogies Longitudinal distance between driving wheel sets in bogie Longitudinal distance between guiding wheel sets in bogie

kg kg N/m N/rad N/m kN∙s/m kN∙s/m kN∙s/m m m m

8000/11740 1700 3e6 11,400 3e6 8 3 30 5.975 1.5 2.49

Table 5 Natural frequencies of the bridge. Modes

Calculation results /Hz

Test results /Hz

Damping ratio (%)

First vertical bending mode First coupled lateral-and-torsional mode Second vertical bending mode Second coupled lateral-and- torsional mode

5.36 2.33 14.88 9.26

5.60 2.27 15.19 9.55

1.7 2 – –

The bridge dynamic displacements and the car body accelerations are drawn in Figs. 13–15 when the vehicle passes through the straight bridge at the speed of 40 km/h, together with those of the field test data. As shown in Figs. 13–15, the dynamic displacements of the bridge structure, the vibration accelerations of the car body and the vibration accelerations of the bridge middle-span in the vertical and lateral directions obtained by the proposed 15

Simulation result Test result

(a)

Lateral displacement (mm)

Vertical displacement (mm)

30

20

10

0

Simulation result Test result 10

5

0

-5

-10 4

8

12

16

4

8

12

16

Time (s)

Time (s) 15

25

Simulation result Test result

Simulation result Test result

(c)

Lateral displacement (mm)

Vertical displacement (mm)

(b)

10

5

0

-5

20

(d)

15 10 5 0 -5

5

10

15

Time (s)

20

4

8

12

16

Time (s)

Fig. 13. Dynamic displacement comparisons of bridge structure: (a) Vertical displacement at the middle of bridge; (b) Lateral displacement at the middle of bridge; (c) Vertical displacement of pier; (d) Lateral displacement of pier.

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0.30

(a)

Simulation result Test result

Lateral acceleration of car body (g)

Vertical acceleration of car body (g)

0.30

0.15

0.00

-0.15

(b)

Simulation result Test result 0.15

0.00

-0.15

-0.30

-0.30 6

8

10

12

14

6

16

8

10

12

14

16

Time (s)

Time (s)

Fig. 14. Vibration acceleration comparisons of car body: (a) Vertical acceleration of the car body; (b) Lateral acceleration of the car body.

0.3

0.4 Simulaition result Test result

Simulaition result Test result

0.3

Acceleration (g)

Acceleration (g)

0.2 0.1 0.0 -0.1

0.2 0.1 0.0 -0.1

-0.2

-0.2

-0.3

-0.3 7

8

9

10

11

Time (s)

12

13

14

15

7

8

9

10

11

12

13

14

15

Time (s)

Fig. 15. Vibration acceleration comparisons of the bridge middle-span: (a) Vertical acceleration; (b) Lateral acceleration.

model are in good agreement with those of the test data in terms of the dynamic amplitude and response waveform. It should be point out that the lateral displacements at the pier section and the bridge middle section obtained by the proposed model are smaller than those of test results by 2.4 mm, 1.1 mm, respectively. This is probably due to the fact that the proposed model neglects the influence of the pile foundation. It is also noteworthy that the lateral displacement at the pier is 9.3 mm larger than the vertical displacement at the pier, showing that the pier has larger lateral bending deformation under vehicle dynamic load, which should be paid more attentions on it for the design of STM system.

3.2.2. Comparison of the dynamic responses in frequency-domain In order to further verify the reliability of the proposed model, the frequency contents of vertical accelerations of the car body and the bridge obtained by numerical simulation also are compared with those of the field test data. Figs. 16,17 present the analysis results when the vehicle passes through the straight bridge at the speed of 40 km/h. As it can be seen, the main peak frequencies of the vertical acceleration of the car-body and the bridge middle-span obtained by the proposed model are in good agreement with those of the test data, indicating the proposed model can acquire reliable results in studying the dynamic interaction of STM system to a certain extent. It is worth pointing out that the peak frequency of 0.46 Hz is caused by the bridge span and the wavelength near 6.6 Hz is close to the bridge stiffener space and the tire circumference, as shown in Fig. 16. As shown in Fig. 17, several obvious peak frequencies can be identified in the frequency content of vertical acceleration of the bridge middle-span, and the main excited frequencies are around 5.4 Hz, 15.8 Hz and 22.7 Hz, respectively. It is worth pointing out that some discrepancies can be found between numerical simulation results and field test results, which may be produced by the factors that the actual vehicle speed is not complete constant under each speed condition in the field test, and the mechanical parameters used in numerical simulation also have some discrepancies with the test vehicle. From the above comparisons with the field test, it can be concluded that the proposed model can well simulate the vehicle-bridge interactions for the STM system, which can be further used to evaluate the safety and comfort assessment of the vehicle and reliability of the bridge structure.

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0.020

0.016 f=5.4 f=7.0

0.015

f=5-7.8 f=1.80 f=2.24 f=12-17

f=1.35

0.010 f=0.44

f=13 f=0.90

0.005

0.000 0.1

Test result

(a)

1 Frequency (Hz)

Vertical acceleration of car body (g)

Vertical acceleration of car body (g)

Simulation result

f=12-18

f=5.3

0.012

f=13.0 f=1.87 f=2.34

0.008

f=5-8 f=0.93 f=0.46

f=1.4

0.004

0.000 0.1

10

(b) f=6.6

1 Frequency (Hz)

10

Fig. 16. Frequency content of car-body vertical acceleration: (a) simulation; (b) field test.

0.020

(a)

Simulation result f=5.3

f=6.9

f=15

0.015 f=21 f=30

0.010

f=7.8

0.005

0.000 0.1

1

10

Vertical acceleration of the bridge (g)

Vertical acceleration of the bridge (g)

0.020

(b)

Test result f=5.4

0.015

f=22.7 f=15.8 f=30.5

0.010

f=6.7 f=7.8

0.005

0.000 0.1

Frequency (Hz)

1

10

Frequency (Hz)

Fig. 17. Frequency content of the vertical acceleration of the bridge middle-span: (a) simulation; (b) field test.

4. Dynamic interaction analysis of STM system 4.1. Dynamic performance between the vehicle and the flexible bridge In order to elucidate the importance of considering the interaction between the vehicle and the flexible bridge for the STM system, a coupled dynamic model is developed based on proposed analysis method, which consists of a detailed FE model of the bridge structure with a length of 375 m. Then, a STM vehicle dynamic model with the rigid bridge also is established by applying fixed constraints to all the nodes of the bridge FE model of the proposed coupled model in ANSYS, whose simulation results are compared with the proposed coupled model with a flexible bridge. The dynamic performance of STM system is investigated when the vehicle passes through a curve bridge with the horizontal radius of 100 m at the speed of 40 km/h, and the detailed bridge route consists of a 60 m long straight line, followed a 60 m long transition curve, followed a 60 m long circle curve with the horizontal radius of 100 m, followed a 60 m transition curve and followed a 60 m long straight line, respectively. The simulation results of the proposed model accounting the bridge as a flexible body are contrasted with those of the vehicle dynamic model which treats the bridge as a rigid body, as shown in Fig. 18. The random irregularity of the running surface of the box-beam bridge is not taken into account in this section. Fig. 18 shows the comparisons of car-body lateral displacement, car-body lateral acceleration, driving tire radial force and guiding tire radial force, respectively, when the vehicle passes through the curve bridge with the horizontal radius of 100 m at the speed 40 km/h. Some apparent discrepancies can be easily distinguished treating the bridge as a flexible body and rigid body, respectively. First, the maximum values of lateral acceleration and displacement obtained by the proposed coupled model with the flexible bridge are larger than those of vehicle dynamic model considering the bridge as a rigid body by around 0.14 m/s2 and 18 mm, respectively. Further, the discrepancies of radial forces of the driving tires and guiding tires are significant as demonstrated in Fig. 18(c) and (d). All tire radial forces considering the bridge as a rigid body are relatively smoother. However, the radial forces of the proposed coupled model with a flexible bridge appear the periodic peaks and troughs within a bridge span, due to the uneven stiffness from the pier to the middle of the bridge. When the vehicle runs

C. Cai et al. / Mechanical Systems and Signal Processing 118 (2019) 388–407

Lateral displacement of car body (mm)

2.0

2

Lateral acceleration of car body (m/s )

402

Rigid bridge Flexible bridge

(a) 1.5 1.0 0.5 0.0 -0.5 -1.0 0

5

10

15

20

50 Rigid bridge Flexible bridge

(b) 0

-50

-100

-150

25

0

5

10

Time (s)

20

25

30

16 Left tire in front of bogie (Rigid bridge) Right tire in front of bogie (Rigid bridge) Left tire in front of bogie (Flexible bridge) Right tire in front of bogie (Flexible bridge)

12

Left tire in front of bogie (Rigid bridge) Right tire in front of bogie (Rigid bridge) Left tire in front of bogie (Flexible bridge) Right tire in front of bogie (Flexible bridge)

(d)

Driving tire force (kN)

(c)

Guiding tire force (kN)

15

Time (s)

8

4

0

25

20

15

10 0

5

10

15

20

25

Time (s)

0

5

10

15

20

25

Time (s)

Fig. 18. Dynamic response comparisons between vehicle dynamic model with rigid bridge and proposed coupled model with flexible bridge: (a) The lateral acceleration of car body; (b) The lateral displacement of car body; (c) The guiding tire radial force; (d) The driving tire radial force.

on the circle curve bridge, the maximum values of the guiding tire forces and driving tire forces obtained by the proposed coupled model with a flexible bridge are larger than those of vehicle dynamic model considering the bridge as a rigid body by 2.4 kN and 2.6 kN, respectively. Meanwhile, the left guiding tire force obtained by the proposed model with a flexible bridge is smaller than that of vehicle dynamic model considering the rigid bridge by 1 kN. It is worth pointing out that the guiding tires have initial compression force for improving the vehicle running stability, as shown in Fig. 18(c). Based on the simulation results of the proposed model with a flexible bridge, it will be possible to appear insufficient compression forces of the guiding tires due to the uneven bridge stiffness. Especially, if the compression force of the guiding tire is too small, one side guiding tire force could reach the value of zero when vehicle passes through the curve bridge, which may affect the running stability and comfort of the vehicle to some extent. Based on the above research results, it is shown that the dynamic interaction between the STM vehicle and the flexible bridge is significant. The proposed coupled model can better simulate the realistic phenomenon of the coupled vibration of the STM system compared with the vehicle dynamic model with rigid bridge. On the one hand, the lateral displacement obtained by the proposed model is obviously larger than that of the vehicle dynamic model with the rigid bridge, therefore, it is more accurate to evaluate safety limit and riding comfort of the vehicle. On the other hand, the proposed coupled model can simulate the variation of compression force of the guiding tire due to the flexible bridge, which will be helpful to suggest a more reasonable value of the initial compression force for the consideration of improving the vehicle running stability. In addition, the driving and guiding tire forces obtained by the proposed coupled model simulation are also closer to the real value, which can provide some theoretical support for the bridge fatigue analysis and reliability evaluation of the STM system. Overall, the proposed coupled model with flexible bridge should be quite necessary for the STM system to investigate the vehicle-bridge coupled dynamic characteristics. 4.2. Dynamic response analyses under different speed conditions To systematically research the dynamic performance of STM vehicle and bridge, the dynamic responses of vehicle and bridge subsystems are extracted by numerical simulation and field test under different speed condition, and the change rules of the dynamic indices alone with the vehicle running speed are also acquired. In the numerical simulation, the random irregularity of the running surface of the box-beam bridge is shown in Fig. 12.

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4.2.1. Dynamic responses of vehicle system In the section, the vehicle vibration levels are evaluated by investigating the dynamic responses of car body under different speed conditions. Fig. 19 presents the acceleration amplitudes of car body under different speed conditions, and the vehicle stabilities are further assessed by using the Sperling index as shown in Fig. 20. As shown in Fig. 19, the vertical and lateral accelerations of car body gradually increase with an increase of the vehicle speed. Under the above described condition, the maximum vertical and lateral accelerations of car body are about 0.24 g and 0.25 g, respectively. These results are in good agreement with those of the test data. As shown in Fig. 20, the vertical and lateral Sperling indices of the car body also increase with an increase of the vehicle speed, and the lateral Sperling indices are larger than the vertical indices of car body. It is shown that the lateral stability of car body is worse than the vertical stability due to the special vehicle structure. When the vehicle speed is 60 km/h, the vertical and lateral Sperling indices of car body can reach up to 2.73 and 2.75, respectively. And for the other cases, the Sperling indices are lower than 2.5. These simulation results show a good agreement with the filed test data. The above research results indicate that the vehicle has good operation stability. 4.2.2. Dynamic responses of bridge For the STM system, the box-beam bridge with an open bottom is quite different from the straddle monorail bridge, the subway bridge and high-speed railway bridge, which has lower vertical and lateral stiffness. So, the STM bridge can produce larger displacement and acceleration compared with the traditional railway bridge. Figs. 21–24 show the dynamic displacements at the bridge middle-span and pier sections, the accelerations at the bridge middle-span section and the dynamic Mises stress at the weak point A (shown in Fig. 5) of the beam running surface varying with the vehicle speed, respectively. The detailed simulation results and test results are listed in Tables 6–8. Obviously, there is a good consistency between the simulation results and the filed test data. The vertical and lateral displacements at the bridge middle-span and pier sections and the dynamic Misses stress at the point A change slightly with an increase of the vehicle speed. Whereas the vertical and lateral accelerations at the bridge middle-span increase significantly

0.3 Simulation results Test results

0.2

0.1

0.0

20

30

Simulation results Test results

(a)

Lateral acceleration of car body(g)

Vertical acceleration of car body(g)

0.3

40

50

0.2

0.1

0.0

60

(b)

20

Vehicle speed (km/h)

30

40

50

60

Vehicle speed (km/h)

Fig. 19. Car-body acceleration varying with speed: (a) Vertical acceleration; (b) Lateral acceleration.

3.0 (a)

Simulation results Test results

Lateral Sperling indices of car body

Vertical Sperling indices of car body

3.0

2.5

2.0

1.5

20

30

40

50

Vehicle speed (km/h)

60

(b)

Simulation results Test results

2.5

2.0

1.5

20

30

40

50

Vehicle speed (km/h)

Fig. 20. Sperling index of car body varying with speed: (a) Vertical Sperling indices; (b) Lateral Sperling indices.

60

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40 (a)

Vertical displacement (Simulation results) Vertical displacement (Test results) Lateral displacement (Simulation results) Lateral displacement (Test results)

30

Dynanic displacement (mm)

Dynanic displacement (mm)

40

20

10

0

20

30

40

50

(b)

Vertical displacement (Simulation results) Vertical displacement (Test results) Lateral displacement (Simulation results) Lateral displacement (Test results)

30

20

10

0

60

20

30

40

50

60

Vehicle speed (km/h)

Vehicle speed (km/h)

Fig. 21. Dynamic displacement of bridge varying with speed: (a) middle-span; (b) pier.

0.20

0.16 0.14 0.12 0.10 0.08

(b)

Simulaition results Test results

0.30

Acceleration (g)

Acceleration (g)

0.18

0.35

(a)

Simulaition results Test results

0.25 0.20 0.15 0.10

20

30

40

50

60

0.05

20

Vehicle speed (km/h)

30

40

50

60

Vehicle speed (km/h)

Fig. 22. Acceleration of bridge middle-span: (a) Vertical acceleration; (b) Lateral acceleration.

0.35 Vateral acceleration Lateral acceleration

Acceleration (g)

0.30 0.25 0.20 0.15 0.10 0.05

20

30

40

50

60

Vehicle speed (km/h) Fig. 23. Comparision of bridge middle-span acceleration.

with an increase of the vehicle speed. As shown in Tables 6–8, under the vehicle dynamic load, the maximum vertical and lateral displacements of the bridge middle-span, and the maximum dynamic Mises stress at the point A is 23 mm, 13.6 mm, and 31 MP, respectively. By calculating the ratio of the dynamic results and static results, it could be found that the dynamic impact coefficient of the bridge dynamic displacement and stress are less than 1.15 under the described condition, showing the dynamic impact between the vehicle and bridge is smaller compared with the traditional railway vehicle-bridge system.

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50 Dynamic Mises stress of the weak node A

Mises stress (MPa)

40

30

20

10

20

30

40

50

60

Vehicle speed (km/h) Fig. 24. Dynamic stress of bridge.

Table 6 Simulation and test results of bridge middle-span displacement. Speed/km/h

Static load 20 30 40 50 60

Displacement amplitudes of bridge middle-span Vertical displacement/mm

Lateral displacement/mm

Simulation

Test

Simulation

Test

20.0 21.8 22.0 22.2 22.6 23.0

20.2 22.2 21.9 22.6 21.9 22.1

12.0 12.0 12.1 12.4 12.6 13.0

12.4 12.8 13.5 13.5 13.6 12.9

Mises stress amplitudes of point A of beam running surface/MPa Simulation

28.1 29.6 29.8 30.2 30.8 31.0

Table 7 Simulation and test results of pier displacement. Speed/km/h

Displacement amplitudes of pier Vertical displacement/mm

Static load 20 30 40 50 60

Lateral displacement/mm

Simulation

Test

Simulation

Test

10.2 10.2 10.3 10.4 10.6 10.6

9.8 10.6 10.7 11.1 10.4 10.3

17.3 17.5 17.9 18.0 18.3 18.6

19.8 20.2 20.7 20.4 20.5 20.2

Table 8 Simulation and test results of bridge middle-span acceleration. Speed/km/h

Acceleration amplitudes of the bridge middle-span Vertical acceleration /g

20 30 40 50 60

Lateral acceleration /g

Simulation

Test

Simulation

Test

0.100 0.116 0.150 0.169 0.180

0.097 0.099 0.135 0.165 0.170

0.100 0.180 0.230 0.280 0.320

0.110 0.168 0.200 0.260 0.310

It can be known from Fig. 23 that the lateral acceleration is larger than vertical acceleration at the bridge middle-span section under each speed condition. This is mainly attributed to the fact that the lateral stiffness of the bridge with an open bottom is smaller than the vertical stiffness. When the vehicle speed is 60 km/h, the vertical and lateral accelerations reach the maximum values, which are about 0.18 g and 0.32 g, respectively.

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5. Conclusion In this paper, a coupled dynamic model of the STM vehicle-bridge system has been developed based on multibody dynamics and FE method. Both the spatial vehicle model and the detailed 3D FE model of the bridge structure have been implemented entirely based on ANSYS platform by ADPL, and the coupled dynamic interaction between the STM vehicle and bridge has been investigated based on theoretical and experimental analyses. The main conclusions are drawn as follows: (1) Dynamic responses obtained by the proposed model are in good agreement with those of the field test data, indicating that the proposed model is reliable to be used for investigating the coupled dynamic characteristics of the STM system. (2) The proposed model can account the uneven stiffness from the bridge pier section to the middle section, which predicts much larger lateral displacement of the car body than that of the dynamic model with the rigid bridge, so, it is more accurate to evaluate running safety and riding comfort of the vehicle. (3) The proposed model can consider periodic variations of the driving and guiding forces due to the flexible bridge, it can better reflect the realistic phenomenon of the coupled vibration of the suspension monorail system compared with the dynamic model considering the bridge as rigid body. This will be helpful to suggest a more reasonable initial compression force of the guiding tire for the consideration of improving the vehicle running stability, and will be more precise for the fatigue analysis and reliability assessment of the bridge structure. (4) The acceleration and the Sperling index of car body increase with an increase of the vehicle speed. The lateral stability of car body is worse than the vertical stability because of the box-beam bridge with an open bottom. Under all test conditions, the maximum vertical and lateral Sperling indices of car body are around 2.73 and 2.75, respectively, indicating that the vehicle has good operation stability. (5) The first-order natural frequencies of the bridge in the vertical and lateral-torsional directions are about 5.60 Hz and 2.27 Hz, respectively. The vertical and lateral accelerations at the bridge middle-span section increases as the vehicle speed increases, which reach up to 0.18 g and 0.32 g, respectively. The dynamic impact coefficient of the bridge displacement and stress are less than 1.15 under different vehicle speed conditions.

Acknowledgements This work was supported by the National Natural Science Foundation of China (51708457, 51778194), the Science and Technology Project of Sichuan Province (Grant No. 2017GZ0082, and No. 2017JY0215), the Program of Introducing Talents of Discipline to Universities (111 Project) (Grant No. B16041), and Cultivation Program for the Excellent Doctoral Dissertation of Southwest Jiaotong University. References [1] S. Grava, Urban Transportation Systems, Choices for Communities, McGraw-Hill, New York, 2003. [2] T. Kuwabara, M. Hiraishi, K. Goda, S. Okamoto, A. Ito, Y. Sugita, New solution for urban traffic: small-type monorail system, Hitachi Rev. 50 (2001) 139– 143. [3] G. Kouroussis, L. Van Parys, C. Conti, O. Verlinden, Prediction of ground vibrations induced by urban railway traffic: an analysis of the coupling assumptions between vehicle, track, soil, and buildings, Int. J. Acoust. Vib. 18 (4) (2013) 163–172. [4] M. Sugawara, Research on urban monorails corresponding to actual demand, J. Jpn. Monorail Assoc. 91 (2000) 2–26. [5] E. Boehm, H. Frisch, The new operating system of the H-train in dortmund, Verkehr und Technik. 47 (10) (1994) 465–470. [6] H.W. Rahier, P. Scharf, Sicherheitstechnische Prüfung der fahrerlosen Kabinenbahn des Flughafens Duesseldorf, Signal und Draht. 94 (10) (2002) 20–22. [7] N. Zhang, H. Xia, Dynamic analysis of coupled vehicle–bridge system based on inter-system iteration method, Comput. Struct. 114 (2013) 26–34. [8] G. Kouroussis, G. Gazetas, I. Anastasopoulos, C. Conti, O. Verlinden, Discrete modelling of vertical track–soil coupling for vehicle–track dynamics, Soil Dyn. Earthq. Eng. 31 (12) (2011) 1711–1723. [9] A. Andersson, A. O’Connor, R. Karoumi, Passive and adaptive damping systems for vibration mitigation and increased fatigue service life of a tied arch railway bridge, Comput.-Aid. Civ. Inf. 30 (9) (2015) 748–757. [10] L. Ling, M. Dhanasekar, D.P. Thambiratnam, Dynamic response of the train–track–bridge system subjected to derailment impacts, Vehicle. Syst. Dyn. 56 (4) (2018) 638–657. [11] J. Kim, J.P. Lynch, Experimental analysis of vehicle–bridge interaction using a wireless monitoring system and a two-stage system identification technique, Mech. Syst. Signal Pr. 28 (2012) 3–19. [12] Y. Liu, Z. Tan, C. Yang, Refined finite element modeling of a damaged bridge with virtual distortion method coupling solid superelement, Mech. Syst. Signal Pr. 93 (2017) 559–577. [13] S.Q. Wu, S.S. Law, Evaluating the response statistics of an uncertain bridge–vehicle system, Mech. Syst. Signal Pr. 27 (2012) 576–589. [14] Z. Chen, Z.P. Xie, J. Zhang, Measurement of vehicle-bridge-interaction force using dynamic tire pressure monitoring, Mech. Syst. Signal Pr. 104 (2018) 370–383. [15] S.H. Zhou, G.Q. Song, R.P. Wang, Z.H. Ren, B.C. Wen, Nonlinear dynamic analysis for coupled vehicle-bridge vibration system on nonlinear foundation, Mech. Syst. Signal Pr. 87 (2017) 259–278. [16] Z. Jin, G. Li, S. Pei, Vehicle-induced random vibration of railway bridges: a spectral approach, Int. J. Rail Transport. 5 (4) (2017) 191–212. [17] P. Antolín, N. Zhang, J.M. Goicolea, H. Xia, M.Á. Astiz, J. Oliva, Consideration of nonlinear wheel–rail contact forces for dynamic vehicle–bridge interaction in high-speed railways, J. Sound Vib. 332 (5) (2013) 1231–1251. [18] X. Lei, N.A. Noda, Analyses of dynamic response of vehicle and track coupling system with random irregularity of track vertical profile, J. Sound Vib. 258 (1) (2002) 147–165. [19] C.W. Kim, M. Kawatani, Effect of train dynamics on seismic response of steel monorail bridges under moderate ground motion, Earthq. Eng. Struct. D. 35 (10) (2006) 1225–1245.

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[20] W.M. Zhai, H. Xia, C.B. Cai, M.M. Gao, X.Z. Li, X.R. Guo, N. Zhang, K.Y. Wang, High-speed train–track–bridge dynamic interactions – Part I: theoretical model and numerical simulation, Int. J. Rail Transp. 1 (2013) 3–24. [21] K. Goda, T. Nishigaito, M. Hiraishi, K. Iwasaki, A curving simulation for a monorail car, Railroad Conference, Proceedings of the 2000 ASME/IEEE Joint, IEEE (2000) 171–177. [22] C.H. Lee, C.W. Kim, M. Kawatani, N. Nishimura, T. Kamizono, Dynamic response analysis of monorail bridges under moving trains and riding comfort of trains, Eng. Struct. 27 (14) (2005) 1999–2013. [23] M. Naeimi, M. Tatari, A. Esmaeilzadeh, M. Mehrali, Dynamic interaction of the monorail–bridge system using a combined finite element multibodybased model, Proceedings of the Institution of Mechanical Engineers, P. I., Mech. Eng. K-J Mul. 229 (2) (2015) 132–151. [24] R. Meisinger, Dynamic analysis of the Dortmund University campus sky train, Nürnberg, Germany, Technische Hochschule Nürnberg Georg Simon Ohm, 2006. [25] R. Meisinger, Analysis of the lateral dynamics of a sky train with periodic track irregularities, in: Proc of the Second International Conference on Dynamics, Vibration and Control, Beijing, China, 2006 [26] Y. Bao, Y. Li, J. Ding, A case study of dynamic response analysis and safety assessment for a suspended monorail system, Int. J. Env. Res. Pub. He. 13 (2016) 1121. [27] H. Pacejka, Tire and Vehicle Dynamics, Elsevier, 2005. [28] C.H. Lee, M. Kawatani, C.W. Kim, N. Nishimura, Y. Kobayashi, Dynamic response of a monorail steel bridge under a moving train, J. Sound Vib. 294 (2006) 562–579. [29] J.D. Zhuang, Advanced Technology of Tire, Beijing Institute of Technology Press, Beijing, 2001. [30] H.B. Pacejka, E. Bakker, The magic formula tyre model, Vehicle. Syst. Dyn. 21 (S1) (1992) 1–18. [31] Baffet G, Charara A, Stéphant J. Sideslip angle, lateral tire force and road friction estimation in simulations and experiments, Computer Aided Control System Design, 2006 IEEE International Conference on Control Applications, 2006 IEEE International Symposium on Intelligent Control, 2006 IEEE. IEEE, 2006, pp. 903–908. [32] J.B. Ma, Q.H. Pu, X. Huo, Vehicle-bridge coupling vibration analysis of PC rail beam of straddle-type monorail transportation, J. Southwest Jiaotong Uni. 44 (6) (2009) 806–811. [33] W.M. Zhai, Two simple fast integration methods for large-scale dynamic problems in engineering, Int. J. Numer. Meth. Eng. 39 (24) (1996) 4199–4214.